Interpolation computing system for automatic tool control



Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 1 ROBERT W. TR l PP,

INVEN TOR.

BY 7 w .4 TTORNEY- R. w. TRIPP 3,066,868 INTERPOLATION COMPUTING SYSTEMFOR AUTOMATIC TOOL CONTROL Dec. 4, 1962 15 Sheets-Sheet 2 Original FiledNov. 15, 1956 ROBERT W. TRIPP,

INVENTOR.

ATTORA/EK Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNOV. 15, 1956 15 811881255118812 5 A TTORNEK Dec. 4, 1962 R. w. TRIPP3,065,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 l5 Sheets-Sheet 4 D 'rAY i AX D AY AY D AX I YR ' +Ror-RXR ROBERT W. TRIPP,

INVENTOR.

AT TORNEK Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 5 DIGITAL COMPUTER FIG. IO FIG. II FIG. l2FIG I3 FIG. l4 FIG.I5 FIG. I6

ROBERT W. TRIPP,

INVENTOR.

A T TORWE I.

Dec. 4, 1962 R. W. TRIPP INTERPOLATION COMPUTING SYSTEM FOR AUTOMATICTOOL CONTROL 15 SheetsSheet 6 3 REF CONTROL EMULTJPLIERI READERDISTRIBUTOR STORAGE A RELAYS l 5 6' x x l a 1 TRANSLATOR Z FEED RATE

222 FEED RATE w convsmen 930 ROBERT w TRIPP, VARIAB1I E EAIN INVFJVTOR.11, cow R FROM TAPE DATA Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 7 MULTIPLIERS 95 AX 6 :322 git "Ki GAIINCONTROL REE -m I7 I?2 AY SLOPE ANGLE REE AIY :21; AY AX, 22

'1 ABSOLUTE vALuE OF SLOPE ANGLE OFCURVE i 10 53b AXAY 223 1 STORAGEREF.

RELAYS 2|4 Z52 A|Y+YR AMPLIFIER REE 24 A|X+XR AMPLIFIER I REF. 3;] 19oFEED SERVO AMPLIFIER m ROBERT WTRIPP;

INVENTOR. @012! ATTORNEK R. w. TRIPP 3,066,858 INTERPOLATION COMPUTINGSYSTEM FOR AUTOMATIC TOOL CONTROL- Dec. 4, 1962 15 Sheets-Sheet 8Original Filed Nov. 15, 1956 ROBERT W. TRIPP,

IN V EN TOR.

A T TOPNE Y.

R. w. TRIPP 3,066,868 INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOLCONTROL Dec. 4, 1962 15 Sheets-Sheet 9 Original Filed Nov. 15, 1956HIHIIIHI wkiian SPEED UP TT R MECHANISM X BINARY GEA INTER PO LATORCONTROL 215 SEQUENCE BRAKE a CLUTCH 4 2 BINARY GEAR MECHANISM AY L l 1Li I J. L L J D UP SOLENOID ACTUATORS ROBERT W. TRIPP,

IN VEN TOR.

ATTORNEK Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet l0 X POSITION sanvo MOTOR A|X+XR g u 60 ISERVO MOTOR 70 66 6| i 1 g RESOLVERS *"37 RESOLVERS L Q i '"T ""7""? 5X] AX 6 D 44) 3 X 13 FEED RATE /43 SERVO ncuousrzn MOTOR sznvo MOTORRESOLVERS Y posmon SERVO MOTOR ROBERT W. TRIPP, INVENTOR.

4 BY W A TTORNEK Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 11 ROBERT W TRIPP,

INVENTOR.

ni {6'- BY A TTORNEK.

Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 12 I O I l 1 2 i g N (T) I N m "a 8 o 2ROBERT W. TRIPP,

INVEN TOR.

4 T TOR/V5 K Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet l3 ROBERT W. TRIPP,

INVIJVTOR.

4 I-r: BY W 5? I70 9 m ATTORNEY.

Dec. 4, 1962 R. w. TRIPP 3,066,863

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 14 9 5I2X l 2 98 256X T"' J. nae. a

OUTPUT s ROBERT w. TRIPP,

mmvroa.

A T TOR/V5 K Dec. 4, 1962 R. w. TRIPP 3,066,868

INTERPOLATION COMPUTING SYSTEM FOR AUTOMATIC TOOL CONTROL Original FiledNov. 15, 1956 15 Sheets-Sheet 15 9 L0 r I0 (\I (SBHONI NI) SlNlOdN33Mi38 BONVLSICI ROBERT W. TRIPP,

INVENTOR.

A TTORNE'K United States Patent ()fil'lC6 3,066,868 Patented Dec. 4,1962 21 Claims. (Cl. 235-197) The invention relates to a computer systemfor an automatic machine tool control which will accept input data indigital form and control the motion of a tool relatively to a Work piecealong a straight path, or along a continuous curve instead of generatinga straight line segmental approximations as heretofore proposed, forexample in the M.I.T. milling machine control. The tool may be eitherthe cutting tool of a milling machine, lathe, profiler or the like, orit may be the scriber or stylus of a drafting or engraving machine forgenerating curves.

In U.S. application, S.N. 557,035 filed Ian. 3, 1956, Patent 2,875,390,February 24, 1959, for Automatic Machine Control Method and System,hereafter referred to as Case 3 there is disclosed and claimed anautomatic machine tool control method and system which will also acceptdigital input data for the purpose above mentioned. Case 3 and otherpatent applications referred to herein are assigned to the assignee ofthe present application.

The objects of the present invention include the features of reducingthe amount of input data required, eliminating much of the effortrequired for programming the portion of the computation carried out on adigital computer, and resolving the instruction to the machine elementsinto a plurality of grades of increments of which the fine increment ispreferably handled by a highly accurate position-measuring transformersuch as the Inductosyn 1 as described in further detail later.

Case 3 and several other pending applications referred to later describeand claim the generation of a complex curve by what is essentiallyextrapolation, in that the various inputs namely, the coordinates of thepoints and slope, curvature and rate of change of curvature areintegrated to direct the path towards the next point on the curve to begenerated. This approach involves digital-toanalog conversion of inputdata into a shaft rotation, resolving the angular position or movementof the shaft into co-function values having the relation of sine andcosine of the shaft angle and integrating the feed rate of the machinedrives along the X and Y axes with such sine and cosine valuesrespectively whereby the machine elements are controlled relatively toeach other in a manner defining the straight line or complex curvedesignated in the input. Depending upon the accuracy of the constantsand of the integration, the line or curve thus generated may or may nothit the next point exactly. Thus there is need to provide for checkingand correcting as described and claimed in US. application S.N. 563,125,now abandoned filed February 2, 1956 for Automatic Digital Machine ToolControl, hereinafter referred to as Case 4.

As compared to Cases 3 and 4, an object of the invention is to simplifythe apparatus required, avoid inaccuracies due to use of mechanicaldevices such as balldisk-cylinder integrators for performingmathematical operations, avoid the necessity for checking thecorrectness of the tool path, and provide a method and apparatus whereinthe correctness of the tool path is inherently checked, while generatingor machining a continuous curve. This is accomplished by generating thecurve by an interpolation method in which the curve is fitted be- 1Registered Trade Mark tween point pairs and goes through each pointwithout errors of integration. The accuracy of the approximation to aspline fit depends upon the separation of the points and upon the natureof the curve as explained in further detail later. The amount of datarequired according to the invention can be reduced by selecting thedistance between data points with due regard to the required curve shapeand the permissible error. The invention also makes it possible tomaintain a constant cutting speed even though each cycle of the feedrate includes point pairs having different separations.

It is possible to obtain the required machine input data with certainrelatively simple arithmetical operations. The specification developsthe required mathematical equations. The only original data needed forthe evaluation are the X and Y coordinates of a suitable number ofpoints. When the equation of the curve is known, it is a very simplematter to compute the necessary input data to the machine, as will behereinafter explained.

A particular advantage of the invention is that the machine can bestopped at any point of its sequence and be started again without lossof accuracy.

According to the invention, a curve is divided into segments, and thecoordinates of each segment are taken as the sum of the coordinates ofpoints along the chord between the point pairs for each segment and thedistance from the chord to the curve, measured perpendicular to thechord. The invention provides method and means for continuouslycomputing this sum in analog form from data of the segment and the chordand data of the required interpolation constants. A further object ofthe invention is to continuously compute from such data the lengths ofthe perpendicular from the chord to the curve, resolve suchperpendicular values into their components along the X and Y axes, addsuch components to the corresponding components of the chord along suchaxes, and control the feed rates of the machine elements along thoseaxes accordingly.

While the invention provides methods and means for generating a straightline, or a succession of straight lines having different slopes, theinvention deals particularly with the generation of complex curves bygenerating successive curved segments. This involves the chord-totangentangle and the length of the chord for each segment and for thegeneration of continuous curves, it also involves preparing theinstruction for the next segment while maintaining the instruction for agiven segment with a quick shift from one instruction to the next. It isa particular object of the present invention to provide method and meansfor these purposes. Further objects of the invention are to provide foradjusting the position of the required curve before starting, to providefor a separate introduction of cutter offset instruction, to provide aconstant cutting rate independent of the length of each successivesegment of the curve to be cut. A further object is to provide for thegeneration of corners including means for stopping the feed rate driveuntil the new slope angle is established, while maintaining the cutterradius instruction active so that at the corner the machine willdescribe an arcuate path having a radius equal to the cutter radiuswhereby the cutter always remains in contact with the corner being cut.Provision is also made for zero offset as described and claimed inapplication S.N. 638,722 filed February 7, 1957, now Patent No.2,950,427, for Zero Oflset for Machine Tool Control. In general, theseobjects and features are accomplished as follows.

The curve is divided into a plurality of successive segments, the lengthof these segments being chosen with due regard for the required accuracyas explained later.

Slope angle-The slope of the path depends on the ratio of the feed ratesalong the X and Y coordinate axes. This ratio is established byextending each of the X and Y drives through a variable speed ratiodevice, here illustrated as an improved digital gear device. The digitalgear device disclosed and claimed herein is binary and, as to itsgeneric aspects, is described and claimed in Case 3.

Distance-The chord of each successive segment of the curve is obtainedas follows. The two gear ratio units referred to above have an inputshaft which is driven through one revolution or an integral number ofrevolutions for each segment, corresponding to a cycle of operation, andthe gear ratio is changed only at the end of a cycle. This results inthe X and Y output drive shafts making a number of revolutions or partsof revolutions proportional to the digital inputs. Thus the totalangular travel of the two output shafts, for the X and Y axes, isproportional to the distance along those axes required to generate thechord of the segment.

Instruction shift.-Storage relays are provided for transfer of the inputdata into the gear ratio units during the cycle, without any pause inthe motion of the output shafts for continuous curves. Also, each of thegear devices is constructed and arranged to maintain a given ratio whilebeing urged to an alternate position corresponding to a change in theratio, such change being prevented, however, until the end of the cycle,where the change is quickly effected. During the cycle, theanalog-computer operates in synchronism with and may be driven by theinput shaft for the gear units so that the computation of the curve data(X and Y components of the perpendicular distance from the chord to thecurve), proceeds in space phase relation with the X and Y feed rateinstructions which determine the length and slope of the chord.

Psiti0n.This is added to the instruction generated by the analogcomputer equipment so that the machine may generate a curve at anydesired point within its capacity.

Cutter ofiseL-Provision is made for computing the absolute value of theslope angle of the curve from the input values of the increments AX andAY, an adjustable tool radius input R being provided to resolve the toolradius R into the increments X and Y for addition to the otherinstructions pertinent to the X and Y axes. This is broadly describedand claimed in U.S. application S.N. 561,769 filed January 27, 1956 forTool Radius Correction Computer as applied to two axes, and is describedand claimed as applied to three axes in S.N. 608,357 filed September 6,1956 for Three Dimensional Tool Radius Correction Computer, and SN.656,692

4 filed May 2, 1957, now abandoned, for Three Dimensional Linear ValueComputer. Use of this feature permits various cutters to be employedwithout altering the input program.

Constant cutting rate.-As successive cycles may involve chords ofdifferent lengths, provision is made for operating the input drive tothe gear ratio units at a speed inversely proportional to the length ofthe chord whereby a substantially constant cutting rate is obtained.

Corner r0utine.As described above, provision is made for reducing thefeed rate to zero at a corner until the instruction of a new slope angleis set in, while maintaining the cutter radius instruction active sothat the cutter always remains in contact with the corner being cut.

Zero 0fiset.Provision is made for locating the program zero as describedwith respect to the machine coordinates so that the part may beprogrammed in advance and the part located on the machine subsequently.The difference between the machine coordinate zero and the partcoordinate zero is zero offset. This is described and claimed in U.S.application S.N. 638,722 filed February 7, 1957, now Patent No.2,950,427 for Zero Offset for Machine T 001 Control.

Sequence.-Concerning the sequence of operation of the digital geardevice in relation to the program advance, the objects of the inventionare to hold the inputs to the differentials in the selected state fromone command to another, change from one state to another at exactly theposition desired, hold the digital input for a short time until theequipment is ready to accept it, hold the inputs energized independentlyof the storage circuits, utilize the storage circuits to hold the nextdata while current data is being used, and to provide a circuit whichmakes it possible to read the punched card or tape at a relatively slowrate and during times when the previous information is being held on theoperating coils, while making it possible to change the state of theoperating coils, and obtain the shaft speed called for by the input,very rapidly and at an accurately chosen time or under accurately chosenconditions.

In prior control systems referred to above, it has been customary tomachine successive straight line segments to produce an approximation ofthe desired surface. A more satisfactory machined part and a very largereduction in the amount of the required input data, as well as increasedflexibility and accuracy are obtained according to the present inventionwherein the inputs include differences of position of points on thesurface and interpolation constants for the surface to be cut, wherebythe method and system of the present invention are capable of machiningcontinuous curves instead of straight line segmental approximations ofthe curve or surface to be cut.

In accordance with the present invention, each successive segment of acurve is generated from the values of (a) the differences between thecoordinates of the end points of the segment, (b) the angle between thechord and the tangent at the cutting point, and (-c) the departure H ofthe curve from the chord, measured perpendicularly to the chord from thecutting point on the curve. It is shown that the value of H is given bythe equation:

and that the value of the angle A is given by the equation:

In the above Equations 1 and 2,

fi starting interpolation angle of segment a =ending interpolation angleof segment where 5 is the angle between the chord and the tangent to thecurve at the first data point 11 of the curve segment and a is the anglebetween the chord and the tangent to the curve at the second data pointN+1 of the curve segment.

D=chord distance between adjacent data points S=distance measured alongthe chord to a perpendicular from the chord to the cutting point on thecurve It is further shown that instantaneous values of the coordinatesof points X, Y, on the path to be followed by the center of the cutterare given by the following equations which appear later as Equations 62and 63:

In Equations 62 and 63,

X =abscissa of the first data point N X=ditference in abscissae betweenadjacent data points X =X component of cutter radius Y =0rdinate of thefirst data point Y=difference in ordinates between adjacent data pointsY =Y component of cutter radius It is also shown that the value of tan xas given in Equation 2 is taken into account in computing the cutteroffset components X and Y A further object of the invent-ion is toprovide a method of computing the necessary input values of u, [3, AXand AY in digital form and continuously computing signal values of H,(Equation 1) and H/D, and also tan 1\, (Equation 2) and from thesesignal values to continuously compute the terms included in Equations 62and 63 and add the terms for each equation as indicated, and control thefeed rate of the machine elements on coordinate X and Y axesaccordingly.

It is shown that in the case where the equation of the curve is known,the values of oz and [3 are readily determined, whereas, if the equationof the curve is not known, the values of 0L and ,8 may be computed byanalogy to a spline fit, from data of the points of preceding andsucceeding segments of the curve.

The above objects are accomplished by employing a precision positionmeasuring transformer to attain high precision, and by employing analogcomputers to provide continuous data of position, to drive servos whichposition the machine element relatively to the work piece to be cut.

While the invention will be described with reference to two orthogonalaxes, referred to as the X and Y axes, with the Z axis used forpositional control only, it will be apparent that the interpolationmethod and means herein described may be extended to apply also to athird axis Z at right angles to the plane of the X and Y axes.

There are several known mathematical interpolation methods by which anequation, or a series of equations, can be obtained, which will closelyapproximate any desired curve. In the preferred mathematical methodherein described, the equations of the curve are not computed orrequired. Instead, the mathematical theory of stress and strain isemployed to compute selected parameters of a spline fit to the requiredcurve, or a very close approximation to such spline fit.

The selected parameters are computed for successive pairs of pointsalong the desired curve as given, or taken sufiiciently close togetherto insure the required accuracy of approximation to the desired curve.The computation for each pair of points is based upon the relativelocations of prior and subsequent points along the curve. So far as isknown, this mathematical method of producing a close approximation to adesired curve has previously not been disclosed.

The equations which are herein developed and employed to give eflect tothe location of prior and subsequent points in determining the path tobe followed between each successive pair of points, are based upon theuse of two prior and two subsequent points. It is to be understood,however, that the mathematical method is equally applicable to singleprior and subsequent points or to three or more prior and subsequentpoints, and that the invention is therefore not limited to anyparticular number of prior and subsequent points.

The selected parameters to be used are computed on a digital computerwhich may be of any suitable type, and which is not claimed as part ofthe invention. These parameters are recorded by the digital-computer inany convenient form, such as punched cards or punched paper tape.

The computed parameters are employed in a combination of electrical,electronic and mechanical components to be described, to cause thecutter or other tool of a milling machine or the like to reproduce thedesired curve upon any desired number of work pieces.

In the illustrative embodiment herein described, the X and Y componentsof the perpendicular distance from the chord between any two points tosuccessive points on the interpolated curve are computed in analog formand added to the successive X and Y components of the chord, to guidethe cutter or other tool along a path which will reproduce theinterpolated curve.

The present application is a division of Ser. No. 622,397, filedNovember 15, 1956 by Robert W. Tripp for Interpolating Method and Systemfor Automatic Machine Tool Control which discloses and claims themachine tool features of the present case. The variable speed drivedisclosed herein is disclosed and claimed in Ser. No. 683,378, filedSeptember 11, 1957, for Variable Speed Drive Interpolation Method andSystem for Automatic Machine Control. The binary gear device disclosedherein is disclosed and claimed in Ser. No. 683,402, now Patent No.2,902,887, filed September 11, 1957, for Binary Gear Device. The lasttwo mentioned applications are further divisions of Ser. No. 622,397.

For further details of the invention reference may be made to thedrawings wherein FIGS. 1, 2, 3, 4a to 4e and FIG. 5 are schematicfigures illustrating a curve and its components involved in mathematicalequations given later in connection with computing the segment of thecurve from certain parameters of the curve, FIGS. 4a to 4e, being usefulin connection with computing the starting and ending chord-to-tangentangles of the curve where the curve function is not known. FIG. 4aillustrates a spline fit through 6 points. FIGS. 4b to 4e illustrate the3-point spline components of the curve in FIG. 4a.

FIG. 6a is a schematic diagram of a portion of the system in FIG. 12,illustrating a resolver with its inputs and outputs for computing theslope angle g5 between the X axis and the tangent at the cutting pointon the curve, FIG. 6b being the corresponding geometrical diagram.

FIG. 7a is a schematic diagram of another portion of the resolverreferred to above, illustrating the inputs AX and AY for computing thechord length D, FIG. 7b being the corresponding geometrical diagram.

FIG. 8a is a schematic diagram of another portion of the resolver deviceof FIGS. 6a and 7a illustrating the tool radius input to be resolvedinto components along the X and Y axes, FIG. 8b being the correspondinggeometrical diagram.

FIG. 9 is a block diagram illustrating how FIGS. 10 to 16 are arrangededge-to-edge to illustrate a complete system capable of operating inaccordance with the method of the present invention, the input of FIG.10 operating through the various computers and controls illustrated inFIGS. 11 to 15, to control the machine elements illustrated in FIG. 16.

FIG.17 is an enlarged view of the zero offset control illustratedschematically in FIG. 16, FIG. 17 illustrating this item for the Xmotor, a similar control, not shown, being provided for the Y motor.

FIG. 18 is a view, partly in section, which may be considered either aplan view or a side view in elevation of the gear mechanism andassociated parts illustrated in FIG. 14 for the X axis, a similar gearmechanism with its associated parts not shown being also provided asshown schematically in the Y axis in FIG. 14.

FIG. 19 is an enlarged sectional view of the gear device of FIG. 18, thesection being taken on line 19-19 of FIG. 20, looking in.the directionof the arrows, and illustrating the controls for reversing the drive.

FIG. 20 is a section taken on line 2020 of FIG. 19.

FIG. 21 is a section taken on the broken line 21-21 of FIG. 19 and showsthe gear 159 which meshes with the gears 134 and 135.

FIG. 22 is a diagram of the linear digital-to-analog converter andmultiplier employed in the systems of FIGS. 10 to 16.

FIG. 23 is a graph illustrating the relation of the distance betweendata points to the departure of the generated curve from a circular arc.

FIG. 24 is an enlarged sectional view, with parts broken away,illustrating the clutch or detent arm for each of the ten binary geardrives to 109 in FIG. 18.

aoeasee specification, with their definitions, as illustrated in FIGS. 1to 8.

X=instantaneous value of the abscissa of the cutter center along themachine X axis Y=instantaneous value of the ordinate of the cuttercenter along the machine Y axis X =abscissa along X axis of data point NY =ordinate along Y axis of data point N X abscissa of the cutting pointalong X axis Y =ordinate of the cutting point along Y axis X =Xcomponent of cutter radius Y =Y component of cutter radius AX=differencebetween the abscissae of adjacent data points AY=difference between theordinates of adjacent data points A X=X component of H (see H below) AY=Y component of H (see H below) H departure of the curve from the chordD, measured perpendicular to D from the point P on the curve H'deflection of a beam anchored at point 1 and bent to pass through point2. (See FIG. 2)

D=chord distance between adjacent data points C=constant S=distancemeasured along the chord D R=cutter radius :810196 angle between thechord D and the X axis 7\=angle between the chord and the tangent at thecutting point P 0=the angle between two successive chords =slope angleof the tangent at P fl =the angle between the chord of a curve segmentN, N+1 and the tangent to the curve segment at N, or, chord-to-tangentangle oc =th8 angle between the chord of a curve segment N-1,

N and the tangent to the curve segment at K, or, chordto-tangent angle:5 as applied to Equations 1 and 2. (See FIG. 3)

AAX: (tan x) AY See FIG. 6

AAY: (tan A) AX See FIG. 6

SIGN CONVENTIONS The sign conventions employed in the followingdiscussion are:

(l) Angles are positive when measured counter-clockwise.

(2) An angle between a chord and a tangent is measured from the chord tothe tangent.

(3) An angle between two chords is measured from the extension of thefirst chord to the second chord.

(4) Distances along chords are taken as positive in the direction ofmotion. Distances normal to chords are taken as positive to the left ofthe direction of motion.

(5) Lengths of chords are taken as positive.

MATHEMATICS OF THE INTERPOLATION METHOD Consideration will now be givento the mathematics of the interpolation method, first for the generalcase where the equation of the curve is not known, and then for the casewhere the equation of the curve is given.

Equation of the curve not known.The general case is the one in which theequation of the curve is not known. In this case, the digital computer2', FIG. 10, computes the lengths D of the chords and the angles 0 fromthe values of X and Y between the chords. Using Equations 35, 36, 37 and38 (see column 10, lines 39-40) to find values of C and C.,, it thencomputes the required values of the chord-to-tangent angles at and [3from Equations 33 and 34. These equations are explained later.

Equations 1 and 2 (column 4, lines 40-44) are those of the requiredparameters of a close approximation to the curve which would be producedby a spline, or a uniform flexible strip which is caused to pass throughthe given points on the curve. The use of such a spline or strip iswell-known in the drafting and lay-out arts. The validity of theequations may be demonstrated in the following way: It a beam 1, 2 (FIG.2), fixed at the point 2, is bent by a force normal to the beam at point1 so that the beam after bending passes through the point 1, itsbehaviour may be analyzed in the following way: there is set up abending moment which varies along the beam, its magnitude at any point Pbeing proportional to the distance S, from the point 1. This is awell-known principle, fundamental in the theory of stress and strain,and presented in any treatise on Strength of Materials. In FIG. 2, and,indeed, in the usual treatment of beams, it is considered immaterial tothe discussion whether the distances involved are measured along thetangent 1, A, along the chord 1, 2 or along the bent beam 1, P, 2itself, since the angle A, 1, 2 is considered to be so small that therelationship Angle=Sine {L -Tangent B is a very close approximation. Inthe operation of the invention herein described, it is necessary thatthe points along the curve to be out be taken sufiiciently closetogether that this relationship is true without sensible error. This isnot a tight restriction since, for reasons of accuracy, points wouldnormally be taken sufiiciently close together that this is the case.

The bending moment of the beam which is proportional in magnitude to thedistance S along the beam (or along the tangent, or the chord), causes achange in the slope of the beam such that the rate of change of slope isproportional to said bending moment. From the principles of analyticalgeometry it is known that the slope of a curve H =f(S) is given by thefirst derivative of H with respect to S, and the rate of change of slopeis given by the second derivative of H with respect to S, or in commonnomenclature, and with the notation of FIG. 2:

=f(S)=Rate of Change of Slope (8) But it is known that the rate ofchange of slope is proportional to S, hence we may write and if a doubleintegration is performed, there results the expression where C and C areconstants of integration, C representing the initial slope and C theinitial deflection of the beam. In the situation represented in FIG. 2,both are Zero, hence H=kS (11) To determine the value of k, We note inFIG. 2 that, when S=D, then H'==D tan fi=DB by the small-angle We areinterested, however, in the value of H, or departure of the curve fromthe chord. Referring to FIG. 2 and using the small-angle approximation:

fl and, since H=(H+H') -H' (l5) from Equations 13, 14 and 15, we have:

H=s/s-s 16) which may be written S3 H=(s- ,)a 7) and the slope of thecurve with respect to the chord may be obtained by taking the derivativeof (17) with respect to S, obtaining dH 3S Now, at the point 2, S=D, andsubstitution in (18) gives dH d8 where a is the chord-to-tangent angleat the point 2.

In the case of the present invention, the beam in question (the assumedspline) is not fixed at either end of a given chord, so that the aboveanalysis is insufficient to completely describe the case. The case maybe described, however, to a close degree of approximation by consideringa set of six (6) points of constraint, two to the left of and two to theright of the chord segment being considered. If this is done, thecondition shown in FIGS. 4a to 4e exists, where the symbolism isconsistent with that adopted in the above discussion.

In accordance with the principle of superposition, there is a unique setof four 3-point beams which determines the shape of the spline throughsix points. As will be explained in more detail presently, thechord-to-tangent angles a and {3 of the 3-point beams are proportionalto the chord lengths, and depend in magnitude upon the angles betweensuccessive chords, since, at each point, the sum of the a componentsplus the sum of the 8 components is equal to the chord-to-chord angle 0.(See Equation 28.) The \sum of a and 18,, at the center of each 3-pointbeam is an angle 6, which in general is not equal to 0, and need not bedetermined.

FIGS. 4b to 4e show the component 3-point splines used to compute the6-point spline 4a. The curvature is zero at the end points of the four3-point beams. Hence, at either side of the mid point of each 3-pointbeam, the ,c-hord-to-tangent angle is twice that at the end of the beam,but is of opposite sign. (See Equation 19.)

The actual values of the chord-to-tangent angles at the chord ends arethe algebraic sums of the component values illustrated in FIGS. 4b to4e, that is to say fin=l an+l bm N= aN+ bN From the theory of beamfiexure, it may be shown that aN N1.N BbN DN,N+1 a-nd in particular, fora chord 3, 4 we have m m (22) This may be written Now, the anglesbetween the chords can be seen by inspection to be equal to the sumof-the actual angles a 10 and [8 at a given point, and since, by theprinciple of superposition, the angles a andB at a point are the sums ofthe respective components of these angles a,, +a and fl -i-fl we maywrite, for the angles between the two chords at the points 3 a=%s+%3la3l bs where the algebraic signs have been appropriately chosen for thedirections in which the angles are indicated. Now, from Equation 19 And,substituting in 28 from Equations 24, 25, 26, 27, 29 and 30, we obtainand the values of ,B and 0a required for use in Equations 1 and 2 forthe same chord are seen to be given by In order to obtain the values ofthe constants of proportionality, C, for the chord 3, 4, it is necessaryto set up the equations for 0 0 0 and 6 in simultaneous form, asfollows:

Thus, information pertaining to all of the six points of constraint, isincluded in Equation 1 which is the ordinate of the curve between thepoints 3 and 4, referred to the chord as the X axis.

Equation'l is derived from Equation 17 as follows:

Referring to FIG. 3, H, is the departure from the chord of the curve (a)defined by 3,.

where H, is the departure from the chord of the curve (12) defined by aThe resultant curve is

